Superresolution in pulse-echo imaging via the use of a transform has been previously described for stabilized asymmetric pulses. Stabilized pulses are those which, when sampled, have an exact inverse filter. Stabilized pulses, in this context, are realizable continuous functions in the axial and transverse directions that when sampled have their Z-transform zeroes lying within the unit circle. This corresponds to inverse filters that are stable because their poles lie within the unit circle, such that they are limited in time with bounded output. Such inverse filters are bounded and well behaved in the presence of noise, and proper design of the stabilized pulse, analyzed with the help of the Z-Transform, can be an important part of a superresolution strategy. However, previous work has only been directed to using an asymmetric pulse shape for generating stabilized pulses. Currently used symmetric functions in ultrasound and other imaging techniques do not have a stable inverse filter. Typical ultrasound pulses used for imaging are functions that, when sampled, have Z-transforms with many zeros on and outside of the unit circle. These produce inverse filters with poles outside of the unit circle, leading to unstable filters.
Accordingly, there is a need in the art for a superresolution strategy related to generating stabilized pulses from symmetric pulse shapes.